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MathGroup Archive 2006

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Re: What's wrong with Integrate [ 1/x, {x,1,Infinity}]?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg66559] Re: [mg66546] What's wrong with Integrate [ 1/x, {x,1,Infinity}]?
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Sat, 20 May 2006 04:46:59 -0400 (EDT)
  • References: <200605190740.DAA12874@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Pavel Pokorny wrote:
>   Dear Mathematica friends
> 
> what is the reason Mathematica 5.2 does not compute 
> 
> In[1]:= Integrate [ 1/x, {x,1,Infinity}]
> 
>                              1
> Integrate::idiv: Integral of - does not converge on {1, Infinity}.
>                              x
> 
>                   1
> Out[1]= Integrate[-, {x, 1, Infinity}]
>                   x

Integrate uses convergence testing heuristics which often correctly 
deduce whether an integral converges or not. But, in the "not" case, 
these methods are not generally competent to assess whether the 
divergence is to a particular directed infinity, or, if so, which one.


> while this works:
> 
> In[2]:= Limit [ Integrate [ 1/x, {x,1,b}], b->Infinity]
> 
> Out[2]= Infinity
> 
> I always thought the first formulation 
> is just a shorthand of the second one,
> being equivalent in its meaning.

Definitely not equivalent, from an operational point of view. The latter 
integral involves a finite, albeit symbolic, value. Finding the limit is 
then a different matter in that no convergence assessment need be done, 
just a limit taken.


Daniel Lichtblau
Wolfram Research



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